Isomorphism Theorems for Groups, Rings, Modules, and Algebras

All flashcards

First isomorphism theorem for groups

For a group homomorphism $f:G\to H$, $G/\ker f\cong\operatorname{im}f$.

Second isomorphism theorem for groups

If $S\leq G$ and $N\trianglelefteq G$, then $(SN)/N\cong S/(S\cap N)$.

Third isomorphism theorem for groups

If $N\leq K\trianglelefteq G$, then $(G/N)/(K/N)\cong G/K$.

Correspondence theorem for groups

Subgroups of $G/N$ correspond to subgroups of $G$ containing $N$.

Congruence in universal algebra

An equivalence relation compatible with every operation of an algebra.

Quiz questions

1. Let $S\leq G$ and $N\trianglelefteq G$. What is the second isomorphism theorem's conclusion?Answer: $(SN)/N\cong S/(S\cap N)$
2. Suppose $N\leq K\trianglelefteq G$. Which isomorphism follows from the third isomorphism theorem?Answer: $(G/N)/(K/N)\cong G/K$
3. For a ring homomorphism $\varphi:R\to S$, what kind of structure is $\ker\varphi$?Answer: A two-sided ideal of $R$
4. For a module homomorphism $\varphi:M\to N$, what is the relationship between its kernel, domain, and image?Answer: $M/\ker\varphi\cong\operatorname{im}\varphi$
5. In universal algebra, what does a congruence allow one to construct?Answer: A quotient algebra whose operations are well-defined on equivalence classes